A Note on Total and Paired Domination
of Cartesian Product Graphs
K. Choudhary
S. Margulies
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Kanpur, India
Department of Mathematics
Pennsylvania State University
State College, PA, U.S.A.
[email protected]
[email protected]
I. V. Hicks
Department of Computational and Applied Mathematics
Rice University
Houston, TX, U.S.A.
[email protected]
Submitted: Sep 9, 2011; Accepted: Jul 27, 2013; Published: Aug 23, 2013
Mathematics Subject Classifications: 05C69
Abstract
A dominating set D for a graph G is a subset of V (G) such that any vertex
not in D has at least one neighbor in D. The domination number γ(G) is the
size of a minimum dominating set in G. Vizing’s conjecture from 1968 states that
for the Cartesian product of graphs G and H, γ(G)γ(H) 6 γ(GH), and Clark
and Suen (2000) proved that γ(G)γ(H) 6 2γ(GH). In this paper, we modify the
approach of Clark and Suen to prove similar bounds for total and paired domination
in the general case of the n-Cartesian product graph A1 · · · An . As a by-product
of these results, improvements to known total and paired domination inequalities
follow as natural corollaries for the standard GH.
1
Introduction
We consider simple undirected graphs G = (V, E) with vertex set V and edge set E. The
open neighborhood of a vertex v ∈ V (G) is denoted by NG (v), and closed neighborhood
by NG [v]. A dominating set D of a graph G is a subset of V (G) such that for all v,
NG [v] ∩ D 6= ∅. A γ-set of G is a minimum dominating set for G, and its size is denoted
γ(G). A total dominating set D of a graph G is a subset of V (G) such that for all v,
NG (v) ∩ D 6= ∅. A γt -set of G is a minimum total dominating set for G, and its size is
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denoted γt (G). A paired dominating set D for a graph G is a dominating set such that
the subgraph of G induced by D (denoted G[D]) has a perfect matching. A γpr -set of G
is a minimum paired dominating set for G, and its size is denoted γpr (G). In general, for
a graph containing no isolated vertices, γ(G) 6 γt (G) 6 γpr (G).
The Cartesian product graph, denoted GH, is the graph with vertex set V (G) ×
V (H), where vertices gh and g 0 h0 are adjacent whenever g = g 0 and (h, h0 ) ∈ E(H), or
h = h0 and (g, g 0 ) ∈ E(G). Just as the Cartesian product of graphs G and H is denoted
GH, the n-product of graphs A1 , A2 , . . . , An is denoted as A1 A2 · · · An , and has
vertex set V (A1 ) × V (A2 ) × · · · × V (An ), where vertices u1 · · · un and v1 · · · vn are adjacent
if and only if for some i, (ui , vi ) ∈ E(Ai ), and uj = vj for all other indices j 6= i.
Vizing’s conjecture from 1968 states that γ(G)γ(H) 6 γ(GH). For a thorough
review of the activity on this famous open problem, see [1] and references therein. In
2000, Clark and Suen [3] proved that γ(G)γ(H) 6 2γ(GH) by a sophisticated doublecounting argument which involved projecting a γ-set of the product graph GH down
onto the graph H. In this paper, we slightly modify the Clark and Suen double-counting
approach and instead project subsets of GH down onto both graphs G and H, which
allows us to prove several theorems/corollaries relating to total and paired domination.
In this section, we state the results, and in Section 2, we prove the results.
Theorem 1. Given graphs A1 , A2 , . . . , An containing no isolated vertices,
γ(A1 )
n
Y
γt (Ai ) 6 nγ(A1 A2 · · · An ) .
i=2
In 2008, Ho [4] proved γt (G)γt (H) 6 2γt (GH), an inequality for total domination
precisely analogous to the Clark and Suen inequality for domination. In this paper, we
extend this result to the n-product case, and then Ho’s inequality becomes a special case
of a more general result.
Theorem 2. Given graphs A1 , A2 , . . . , An containing no isolated vertices,
n
Y
γt (Ai ) 6 nγt (A1 A2 · · · An ) .
i=1
In 2007, Bres̆ar, Henning and Rall [2] proved that γpr (G)γpr (H) 6 8γpr (GH), and in
2010, Hou and Jiang [5] proved that γpr (G)γpr (H) 6 7γpr (GH). We extend these results
to the n-product case, and attain an improvement to these inequalities as a corollary:
Theorem 3. Given graphs A1 , . . . , An containing no isolated vertices,
n
Y
γpr (Ai ) 6 2n−1 (2n − 1)γpr (A1 · · · An ) .
i=1
Corollary. Given graphs G and H containing no isolated vertices,
γpr (G)γpr (H) 6 6γpr (GH) .
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2
2
Main Results
We begin by introducing some notation which will be utilized throughout the proofs in
this section. Given S ⊆ V (A1 · · · An ), the projection of S onto graph Ai is defined as
ΦAi (S) = {a ∈ V (Ai ) | ∃ u1 · · · un ∈ S with a = ui } .
We partition the set of edges E(A1 · · · An ) into n sets. Thus, we define Ei to be
n
o
Ei = u1 · · · un , v1 · · · vn | (ui , vi ) ∈ E(Ai ), and uj = vj , for all other indices j 6= i .
An edge e ∈ Ei is said to be an Ei -edge. For u ∈ V (A1 · · · An ), the i-neighborhood of
u is defined as follows:
n
o
NAi (u) = v ∈ V (A1 · · · An ) | v and u are connected by Ei -edge .
Finally, we present a proposition utilized throughout our proofs. Although the more
general n-dimensional case stated in Prop. 2 is the proposition referenced within the
proofs, we begin by separately stating the 2-dimensional case to clarify the overall idea.
Proposition 1. Let M be a matrix containing only 0/1 entries. Then at least one of the
following two statements are true:
(a) each column contains a 1,
(b) each row contains a 0 .
Prop. 1 refers only to binary matrices, or matrices containing only 0/1 entries. Prop.
2 refers to n-ary matrices, or (in this case) matrices containing only entries in {1, . . . , n}.
Furthermore, Prop. 1 refers only to two-dimensional matrices, or d1 × d2 matrices M .
Prop. 2 refers to n-dimensional matrices, or d1 × d2 × · · · × dn matrices M .
Definition 1. Let M be a d1 × d2 × · · · × dn , n-ary matrix. Then M is a j-matrix if there
exists a j ∈ {1, . . . , n} (not necessarily unique), such that each of the d1 × · · · × dj−1 ×
1 × dj+1 × · · · × dn submatrices of M contains an entry with value j.
Proposition 2. Every d1 × d2 × · · · × dn , n-ary matrix M is a j-matrix for some j ∈
{1, . . . , n} (not necessarily unique).
Note that, given any d1 × d2 × · · · × dn matrix, there are dj submatrices of the form
d1 × · · · × dj−1 × 1 × dj+1 × · · · × dn . Following standard Matlab notation, we will denote
such a submatrix as M [:, . . . , :, ij , :, . . . , :] with 1 6 ij 6 dj .
Example 1. Here we see a 4 × 3 × 3, 3-ary (the entries are contained in the set {1, 2, 3})
matrix M . Using the notation specified above, M [2, :, :] is denoted by the gray shaded
region and M [:, 1, :] is denoted by yellow shaded region. M is both 1-matrix and 2-matrix,
but not a 3-matrix.
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2
3
1
2
1
2
1
1
1
2
3
3
2
2
1
2
2
1
2
2
1
1
3
2
1
3
3
Proof. For a pigeon-hole principle style proof, let M be a d1 × d2 × · · · × dn n-ary matrix
which is not a j-matrix for 1 6 j 6 n − 1. We will show that M is an n-matrix.
Consider j = 1. Since M is not a 1-matrix, there exists at least one 1×d2 ×d3 ×· · ·×dn
submatrix that does not contain a 1. Without loss of generality, let M [i1 , :, . . . , :] with
1 6 i1 6 d1 be such a matrix. Next, consider j = 2. Since M is also not a 2-matrix,
let M [:, i2 , :, . . . , :] with 1 6 i2 6 d2 be a d1 × 1 × d3 × · · · × dn submatrix that does not
contain a 2. Therefore, M [i1 , i2 , :, . . . , :] is a 1 × 1 × d3 × · · · × dn submatrix that contains
neither a 1 nor a 2. We continue this pattern for 1 6 j 6 n − 1. Since M is not a j-matrix
for 1 6 j 6 n − 1, let M [i1 , . . . , in−1 , :] be the 1 × 1 · · · 1 × dn submatrix containing no
elements in the set {1, · · · , n − 1}. Therefore, for all 1 6 i 6 dn , M [i1 , . . . , in−1 , i] = n,
all of the d1 × · · · × dn−1 × 1 submatrices of M contain an entry with value n. Thus, M
is an n-matrix.
We now present the proofs of Theorems 1 through 3.
2.1
Proof of Theorem 1
1
Proof. Let {u11 , . . . , u1γ(A1 ) } be a γ-set of A1 . Partition V (A1 ) into sets D11 , . . . , Dγ(A
1)
such that u1j ∈ Dj1 ⊆ NA1 [u1j ] for 1 6 j 6 γ(A1 ). Having partitioned V (A1 ) based on
a minimum dominating set, we will now partition each of V (A2 ), . . . , V (An ) based on a
minimum total dominating set. For i = 2, . . . , n, let {ui1 , . . . , uiγt (Ai ) } be a γt -set of Ai , and
D1i , . . . , Dγi t (Ai ) be the corresponding partitions of V (Ai ) such that Dji ⊆ NAi (uij ). Note
that this implies uij ∈
/ Dji .
1
Now let Q = {D11 , . . . , Dγ(A
} × {D12 , . . . , Dγ2t (A2 ) } × · · · × {D1n , . . . , Dγnt (An ) }. Then Q
1)
n
Y
forms a partition of V (A1 · · · An ) with |Q| = γ(A1 )
γt (Ai ).
i=2
Let D be a γ-set of A1 · · · An . Then, for each u ∈ V (A1 · · · An ) not in D, there
exists an i such that NAi (u) ∩ D is non-empty. Based on this observation, we define an
n-ary |V (A1 )| × · · · × |V (An )| matrix F such that:
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4
(
1
F (u1 , . . . , un ) =
imin
if (u1 · · · un ) ∈ D , else
where imin = min{i | NAi (u1 · · · un ) ∩ D 6= ∅} .
Observe that F (u1 , . . . , un ) = 1 has two meanings: either (u1 · · · un ) ∈ D or (u1 · · · un ) is
dominated an an A1 -edge.
For j = 1, . . . , n, let dj ⊆ Q be the set of the elements in Q such that the corresponding
submatrices of F are j-matrices. By Prop. 2, each element of Q belongs to at least one
n
n
Y
X
dj -set. Then, γ(A1 )
γt (Ai ) 6
|dj |.
i=2
j=1
Claim 1. |d1 | 6 |D|.
Proof. Similar to Q, let B = {D12 , . . . , Dγ2t (A2 ) }×· · ·×{D1n , . . . , Dγnt (An ) }. For convenience,
n
Y
denote B as {B1 , . . . , B|B| }, where |B| =
γt (Ai ).
i=2
For p = 1, . . . , |B|, let Zp = D ∩ (A1 × Bp ), and
Sp = Dx1 | the submatrix of F determined by Dx1 × Bp is a 1-matrix,
with x ∈ {1, . . . , γ(A1 )} .
Note that if Dx1 × Bp is a 1-matrix for some x ∈ {1, . . . , γ(A1 )} and p ∈ {1, . . . , |B|}, then
for each w ∈ Dx1 , the submatrix of F corresponding to {w} × Bp contains a 1. Therefore,
{w} × Bp contains a vertex in D or a vertex that is dominated by an A1 -edge. In either
case, {w}×Bp contains a vertex that is dominated by Zp . Thus, the projection of Dx1 ×Bp
on A1 (i.e. Dx1 ) is dominated by the projection of Zp on A1 .
We now claim that for p = 1, . . . , |B|, |Sp | 6 |Zp |. Let Sp = {Di11 , Di12 , . . . , Di1t } and
let ΦA1 (Zp ) be the projection of Zp on A1 . Then, ΦA1 (Zp) dominates ∪tx=1 Di1x , and
for
1
1
i ∈
/ {i1 , i2 , . . . , it }, u1i dominates
D
.
Thus,
Φ
(Z
)
∪
u
|
i
∈
/
{i
,
i
,
.
.
.
,
i
}
A1
p
1 2
i
i t 1 is a
1
dominating set
/ {i1 , i2 , . . . , it } > γ(A1 ), and ui | i ∈
/
for A1 . Now, ΦA1 (Zp ) ∪ ui | i ∈
{i1 , i2 , . . . , it } = γ(A1 ) − t. Thus, |ΦA1 (Zp )| > t. Hence, |Sp | = t 6 |ΦA1 (Zp )| 6 |Zp |.
|B|
|B|
X
X
Finally, we see that |d1 | =
|Sp | 6
|Zp | = |D|.
p=1
p=1
Claim 2. For j = 2, . . . , n, |dj | 6 |D|.
Proof. We prove here that |dn | 6 |D|, but a similar proof can be performed for any other
1
j > 2. Let B = {D11 , . . . , Dγ(A
} × {D12 , . . . , Dγ2t (A2 ) } × · · · × {D1n−1 , . . . , Dγn−1
}. For
1)
t (An−1 )
(n−1)
convenience, denote B as {B1 , . . . , B|B| }, where |B| = γ(A1 )
Y
γt (Ai ).
i=2
For p = 1, . . . , |B|, let Zp = D ∩ (Bp × An ), and
Sp = Dxn | the submatrix of F determined by Bp × Dxn is an n-matrix,
with x ∈ {1, . . . , γt (An )} .
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5
Note that if Bp × Dxn is a n-matrix for some p ∈ {1, . . . , |B|} and x ∈ {1, . . . , γt (An )},
then, for each w ∈ Dxn , the submatrix of F corresponding to Bp × {w} contains an n.
Therefore, each Bp × {w} contains a vertex that is dominated by an An -edge. Therefore,
each vertex in the projection of Bp × Dxn on An (i.e., each vertex w ∈ Dxn ⊆ V (An )) is
dominated by a vertex from ΦAn (Zp ), other than itself. In other words, the projection of
Bp × Dxn on An (i.e. Dxn ) is non-self-dominated by the projection of Zp on An .
We now claim that for p = 1, . . . , |B|, |Sp | 6 |Zp |. Let Sp = {Din1 , Din2 , . . . , Dint }
t
n
and let ΦAn (Zp ) be the projection
∪x=1 Dix is non-self-dominated
n of Zp on An . Then,
/ {i
is a
total dominating set ofAn .
by ΦAn (Zp ), and ΦAn (Zp ) ∪ ui | i ∈
1 , i2 , . . . , it }
n
/ {i1 , i2 , . . . , it } =
/ {i1 , i2 , . . . , it } > γt (An ), and uni | i ∈
Now, ΦAn (Zp ) ∪ ui | i ∈
γt (An ) − t. Thus, |ΦAn (Zp )| > t. Hence, |Sp | = t 6 |ΦAn (Zp )| 6 |Zp |. Finally, we see that
|B|
|B|
X
X
|dn | =
|Sp | 6
|Zp | = |D|.
p=1
p=1
To conclude the proof, we observe that
γ(A1 )
n
Y
γt (Ai ) 6
i=2
n
X
|dj | 6 n|D| = nγ(A1 · · · An ) .
j=1
We conclude this section with the following corollary.
Corollary. Given graphs G and H containing no isolated vertices,
max{γ(G)γt (H), γt (G)γ(H)} 6 2γ(GH) .
2.2
Proof of Theorem 2
Proof. For i = 1, . . . , n, let {ui1 , ..., uiγt (Ai ) } be a γt -set of Ai , and D1i , . . . , Dγi t (Ai ) be the
corresponding partitions of V (Ai ) such that Dji ⊆ NAi (uij ).
Let Q = {D11 , . . . , Dγ1t (A1 ) } × · · · × {D1n , . . . , Dγnt (An ) }. Then Q forms a partition of
n
Y
V (A1 · · · An ) with |Q| =
γt (Ai ).
i=1
Let D be a γt -set of A1 · · · An . Then, for each u ∈ V (A1 · · · An ), there exists
an i such that NAi (u) ∩ D is non-empty. Based on this observation we define an n-ary
|V (A1 )| × · · · × |V (An )| matrix F such that:
F (u1 , . . . , un ) = min{i | NAi (u1 · · · un ) ∩ D 6= ∅} .
For j = 1, . . . , n, let dj ⊆ Q be the set of the elements in Q which are j-matrices. By
n
n
Y
X
Prop. 2, each element of Q belongs to at least one dj -set. Then,
γt (Ai ) 6
|dj |.
i=1
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j=1
6
Claim 3. For j = 1, . . . , n, |dj | 6 |D|.
Proof. The proof is copy of proof of Claim 2, except that here B = {D11 , . . . , Dγ1t (A1 ) } ×
(n−1)
··· ×
{D1n−1 , . . . , Dγn−1
},
t (An−1 )
and |B| =
Y
γt (Ai ).
i=1
Thus we have,
n
Y
i=1
γt (Ai ) 6
n
X
|dj | 6 n|D| = nγt (A1 · · · An ).
j=1
We conclude by observing that Ho’s inequality follows as a corollary to Thm. 2.
Corollary (Ho [4]). Given graphs G and H containing no isolated vertices,
γt (G)γt (H) 6 2γt (GH) .
2.3
Proof of Theorem 3
Proof. For i = 1, . . . , n, let ki = γpr (Ai )/2, and {xi1 , y1i , . . . , xiki , yki i } be a γpr -set of Ai ,
where for each Ai and 1 6 j 6 ki , (xij , yji ) ∈ E(Ai ). Partition each V (Ai ) into sets
D1i , . . . , Dki i , such that {xij , yji } ⊆ Dji ⊆ NAi [xij ] ∪ NAi [yji ] for 1 6 j 6 ki .
Let Q = {D11 , . . . , Dk11 } × · · · × {D1n , . . . , Dknn }. Then Q forms a partition of
n
n
Y
1 Y
V (A1 · · · An ) with |Q| =
γpr (Ai )/2 = n
γpr (Ai ).
2 i=1
i=1
Let D be a γpr -set of A1 · · · An . Then, for each u ∈ V (A1 · · · An ), there exists
an i such that NAi (u) ∩ D is non-empty. Based on this observation, we define n different
matrices F i with i = 1, . . . , n, where each of the n matrices is an n-ary |V (A1 )| × · · · ×
|V (An )| matrix F i such that:
(
i
if u1 · · · un ∈ D , else
F i (u1 , . . . , un ) =
jmin where jmin = min{ j | NAj (u1 · · · un ) ∩ D 6= ∅} .
Thus, each of the n matrices F i with i = 1, . . . , n differs only in the entries that correspond
to vertices in the paired dominating set D.
For j = 1, . . . , n and i = 1, . . . , n, let dij ⊆ Q be the set of the elements in Q which
are j-matrices in the matrix F i . By Prop. 2, each element of Q belongs to at least one
dij -set for each i = 1, . . . , n. Now, if an element q ∈ Q belongs to the dij -set, then q also
belongs to the djj -set. To see this, if Mi and Mj are the submatrices determined by q with
respect to the matrices F i and F j , respectively, then all the entries that do not match in
Mi and Mj have value j in Mj . Thus, each q ∈ Q belongs to at least one dii -set for some
n
n
X
1 Y
γpr (Ai ) 6
|dii |.
i ∈ {1, . . . , n}. Then, n
2 i=1
i=1
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7
Similar to Q, let B = {D11 , . . . , Dk11 } × · · · × {D1n−1 , . . . , Dkn−1
}. For convenience, we
n−1
n−1
n−1
Y
1 Y
denote B as {B1 , . . . , B|B| }, where |B| =
γpr (Ai )/2 = n−1
γpr (Ai ). Since D is a
2
i=1
i=1
γpr -set, the subgraph of A1 · · · An induced by D has a perfect matching. Let
Di = {u ∈ D | the matching edge incident to u is in Ei } .
Then, D can be written as the disjoint union of the subsets Di . For p = 1, . . . , |B| and
i = 1, . . . , n, let Zpi = Di ∩ (Bp × An ), and
Sp = Dxn | the submatrix of F n determined by Bp × Dxn is an n-matrix,
with x ∈ {1, . . . , kn } .
Then,
|dnn |
=
|B|
X
|Sp | .
p=1
Claim 4. For p = 1, . . . , |B|, 2|Sp | 6 2|Zp1 |+ · · · + 2|Zpn−1 |+|Zpn |.
Proof. Let Sp = {Djn1 , Djn2 , . . . , Djnt }, and for j = 1, . . . , n, let Vj = ΦAn (Zpj ). Note that
/ {j1 , j2 , . . . , jt } .
/ {j1 , j2 , . . . , jt } ∪ yjn | j ∈
|Vj | 6 |Zpj |. Additionally, let C = xnj | j ∈
Let M be the matching on Vn ∪C formed by taking all of the {xnj , yjn } edges induced by
the vertices in C, and then adding the edges from a maximal matching on the remaining
unmatched vertices in Vn . Then, E = V1 ∪ · · · ∪ Vn ∪ C is a dominating set of An with M
as a matching. Let M1 = V (M ) and M2 = (Vn ∪ C)\M1 . We note that M1 consists of all
the vertices in C plus the matched vertices from Vn , and M2 contains only the unmatched
vertices from Vn .
In order to obtain a perfect matching, we recursively modify E by choosing an unmatched vertex a in E, and then either matching it with an appropriate vertex, or removing it from E. Specifically, if NAn (a)\V (M ) is non-empty, there exists a vertex
a0 ∈ NAn (a)\V (M ) such that we can add a0 to E and (a, a0 ) to the matching M . Otherwise, a is incident on only matched vertices, and we can safely remove it from E without
altering the fact that E is a dominating set.
Our recursively modified E (denoted by Erec ) is now a paired dominating set of An .
Furthermore, in the worst case, we have doubled the unmatched vertices from Vn , and
also doubled the vertices in V1 , . . . , Vn−1 . Thus,
2kn 6 |Erec | 6 2|V1 |+ · · · + 2|Vn−1 |+|M1 |+2|M2 | .
This implies that 2kn −|C| 6 2|V1 |+ · · ·+2|Vn−1 |+|Zpn |. Since 2kn −|C| = 2|Sp |, therefore,
2|Sp | 6 2|V1 |+ · · · + 2|Vn−1 |+|Zpn | 6 2|Zp1 |+ · · · + 2|Zpn−1 |+|Zpn | .
Using Claim 4, we now see
2
|B|
X
p=1
|B| |B|
n
X
X
X
j
n
|Sp | 6
2
|Zp | − |Zp | = 2|D| −
|Zpn | = 2|D| − |Dn | .
p=1
j=1
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p=1
8
Therefore, 2|dnn | 6 2|D| − |Dn |. Similarly, we can show that 2|dii | 6 2|D| − |Di | for
i = 1, . . . , n. To conclude the proof, we see
n
1 Y
2n−1
γpr (Ai ) = 2(k1 · · · kn ) 6 2
i=1
n
X
|dii | 6 2n|D| −
n
X
i=1
n
Y
|Di | = (2n − 1)|D| ,
i=1
γpr (Ai ) 6 2n−1 (2n − 1)γpr (A1 · · · An ) .
i=1
We conclude this section by observing that an improvement to the Hou-Jiang [5]
inequality γpr (G)γpr (H) 6 7γpr (GH) follows as a corollary:
Corollary. Given graphs G and H containing no isolated vertices,
γpr (G)γpr (H) 6 6γpr (GH) .
Acknowledgments
The authors would like to acknowledge the support of NSF-CMMI-0926618, the Rice
University VIGRE program (NSF DMS-0739420 and EMSW21-VIGRE), and the Global
Initiatives Fund (Brown School of Engineering at Rice University), under the aegis of
SURGE (Summer Undergraduate Research Grant for Excellence), a joint program with
the IIT Kanpur and the Rice Center for Engineering Leadership. Additionally, the authors
acknowledge the support of NSF DSS-0729251, DSS-0240058, and the Defense Advanced
Research Projects Agency under Award No. N66001-10-1-4040. Finally, the authors
would like to thank the anonymous referees for their careful reading and insightful comments which greatly simplified the presentation of the proof.
References
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